ICML 2026 Papers

Existing neural combinatorial optimization (CO) solvers often rely heavily on expensive labeled data and additional post-processing to produce feasible solutions. Research into mixed integer linear programs (MILPs) is particularly limited due to the lack of effective heuristics for feasibility and the challenge of modeling mixed-type variables for neural solvers. To address these issues, we propose a novel unsupervised Langevin sampler for solving MILPs. Our framework learns only integer variables, while continuous variables are solved using an exact linear programming solver, thus isolating the combinatorial hardness of the problem and avoiding unnecessary modeling complexity. The sampler is based on Langevin dynamics and incorporates both objective optimization and constraint satisfaction into a unified energy function, enabling the model to jointly learn feasibility and optimality. Experiments demonstrate that our method achieves 100\% feasibility without expensive post-processing and matches or outperforms supervised solvers on benchmark datasets, highlighting its effectiveness and scalability. Our code can be found at https://github.com/CindyH1103/UNLS4MILP.

In high-dimensional Ising model estimation, target sample sizes are often limited, and effectively using auxiliary binary datasets of unknown relevance remains challenging. To address this, we propose Trans-Ising, a transfer learning method that combines a loss-based source screening rule with a two-stage estimation procedure. The method first identifies informative auxiliary sources using held-out target pseudolikelihood to prevent negative transfer. It then computes an initial estimator via pooled nodewise $\ell_1$-regularized logistic regression, followed by a target-only correction step using a folded-concave penalty. Theoretically, we establish fixed-node $\ell_2$ and $\ell_1$ error bounds, exact graph selection consistency, and the conditional consistency of the screening rule. Through extensive simulations and real-data analyses, we demonstrate that Trans-Ising achieves lower estimation errors than both target-only estimation and naive data pooling.

Compositional data analysis has gained increased attention due to the widespread occurrence of simplex-valued data, including microbiome data and financial portfolios. Existing compositional two-sample tests often require $\log$-transformations and only detect mean differences, motivating the need for a more general framework without relying on $\log$-based methods. There is a close connection between compositional data and directional statistics, and we construct a unified non-parametric two-sample test framework. Our work is based on a studentized energy statistic constructed from spherical harmonics theory over a fixed dimensional underlying space, incorporating U-statistics theory and recent developments of studentization for both compositional and directional data. We establish asymptotic normality for our spherical harmonics based test statistics, thus avoiding the need for permutation tests or bootstrap procedures. Our proposed framework sheds new light on the connections between Non-Euclidean data analysis and classical asymptotic high-dimensional data techniques.

Building on recent formalizations of root cause analysis for rare events (“outliers”) in structural equation models, we propose a formal definition of a causal pathway and discuss its testable implications. We identify conditions under which these implications depend only on a causal abstraction defined by the pathway of rare events, rather than on the full causal graph of the underlying system. Accordingly, we introduce an abstraction of causal structure to pathways of rare events that bridges simple verbal causal explanations and detailed causal modeling.

Kernel methods are powerful tools in statistical learning, but their cubic complexity in the sample size $n$ limits their use on large-scale datasets. In this work, we introduce a scalable framework for kernel regression with $\mathcal{O}(n \log n)$ complexity, and designed to fully leverage GPU acceleration. The approach is based on a Fourier representation of kernels combined with non-uniform fast Fourier transforms (NUFFT), enabling exact, fast, and memory-efficient computations. We instantiate our framework in three settings: Sobolev kernel regression, physics-informed regression, and additive models. The proposed estimators are shown to achieve minimax convergence rates, consistent with classical kernel theory. Empirical results demonstrate that our methods can process up to tens of billions of samples within minutes, providing both statistical accuracy and computational scalability. These contributions establish a flexible approach, paving the way for the routine application of kernel methods in large-scale learning tasks, whenever the kernel norm can be efficiently expressed in Fourier space and the ambient dimension $d$ is small. Although the theoretical framework is valid in all dimensions $d$, the fast Sobolev regression package implemented in the paper relies on the CufiNUFFT package, which currently scales exponentially in $d$ and is limited to $d \leq 3$. For similar reasons, the additive model package is only implemented for simple effects without interaction.

Differentiable logic gate networks map directly to gate-level circuits, enabling ultra-low-latency inference, yet their logic footprint often exceeds FPGA capacity budgets. Tightly fitting a trained model to a target FPGA requires a post-training mechanism to trade off network complexity and accuracy—analogous to pruning in standard neural networks. To this end, we introduce *unit tying*: a simplification that forces selected gates to constants (0 or 1), enabling constant propagation and downstream logic elimination. However, we observe that naively extending pruning criteria to logic networks is unreliable under such near-discrete modifications. We therefore propose a two-stage algorithm for unit tying: (i) a fast Gauss–Newton screening step under a teacher-referenced logit-distortion objective that constructs a high-recall overshoot set and (ii) a refinement step that corrects approximation and interaction-driven errors using a small number of finite-difference evaluations. On CIFAR-10 and MNIST, our method consistently improves the accuracy–area trade-off over common saliency baselines, yielding substantial post-synthesis LUT reductions of up to 48% on CIFAR-10 and 43% on MNIST, with modest accuracy degradation.

Radio Frequency Fingerprint Identification (RFFI) is a foundational pillar of physical-layer security, providing unclonable identity authentication and lightweight defense mechanisms for zero-trust wireless networks. Its practical deployment, however, often occurs in a source-free open-world (SF-OW) setting, characterized by a continuous influx of unregistered devices and privacy constraints that preclude the retention of historical data. In this paper, we formalize SF-OW RFFI task, which manifests a severe stability-plasticity dilemma: intrinsic signal similarity confuses new classes, while source absence precipitates catastrophic forgetting. To address this, we propose Incremental Orthogonal ETF (IO-ETF), a novel neural collapse-inspired framework utilizing output geometry to actively induce parameter separation and isolation. We further devise a Triple-Level Geometric Alignment (TLGA) strategy via semantic optimal transport, manifold progressive anchoring, and reliable subspace retention to stably align unlabeled streams to this geometric skeleton. Experiments on benchmarks demonstrate a superior trade-off between old-class retention and new-class discovery, offering a promising solution for secure access in dynamic networks.

Distributed stochastic optimization intertwines (i) stochastic gradient noise, (ii) communication compression, and (iii) adaptive/normalized updates. While each factor has been studied in isolation, their joint effect under realistic assumptions remains poorly understood. In this work, we develop a unified theoretical framework for Distributed Compressed SGD (DCSGD) and its sign variant Distributed SignSGD (DSignSGD) under the recently introduced $(L_0, L_1)$-smoothness condition. From a conceptual perspective, we show that the first- and second-order modified equations from the literature do not accurately model the discrete-time step-size/stability restrictions, especially under $(L_0,L_1)$-smoothness. From a technical perspective, we propose new first-order SDEs by carefully incorporating curvature-dependent terms into their drift: This helps capture the fine-grained relationship between learning rate restrictions, gradient noise, compression, and the geometry of the loss landscape. Importantly, we do so under general gradient noise assumptions, including heavy-tailed and affine-variance regimes, which extend beyond the classical bounded-variance setting. Our results suggest that normalizing the updates of DCSGD emerges as a natural condition for stability, with the degree of normalization precisely determined by the gradient noise structure, the landscape’s regularity, and the compression rate. In contrast, DSignSGD converges even under heavy-tailed noise with standard learning rate schedules. Together, these findings offer both new theoretical insights and perspectives, and practical guidance.

As Large Language Model (LLM) ecosystems grow, routing queries to the most suitable model in a diverse pool has become a critical strategy for building efficient and high-performing AI systems. A common approach is to train a supervised router; however, this requires vast, expensive human-annotated preference data and creates models that are notoriously brittle, failing to generalize when faced with inevitable distribution shifts in user queries. Consequently, developing robust, unsupervised routing methods that adapt without retraining is a crucial research frontier. This challenge is severely amplified by Large Reasoning Models (LRMs), which introduce a dual problem for any label-free method: their outputs have a causal thinking $\to$ answer structure that must be modeled, and a structural imbalance where long reasoning text can dominate the final answer signal. We introduce ReasoningRouter, a novel framework that resolves these issues with a length-balanced embedding strategy and a probabilistic model capturing the thinking-to-answer dependency. The proposed Causal Triangulation Property enables the label-free estimation of component qualities and their causal link. Beyond competitive routing accuracy, ReasoningRouter offers unprecedented insights into model behavior, enabling separate quality assessment of reasoning and answer components while maintaining computational efficiency.

History-dependent sampling can reduce long-run Monte Carlo variance by discouraging redundant revisits, but existing schemes typically encode history through empirical measure on finite state spaces, which is infeasible in high-dimensional discrete configuration spaces or ill-posed in continuous domains. We propose *Score-Repellent Monte Carlo* (SRMC) framework that summarizes trajectory history by a running average of score evaluations in $\mathbb{R}^d$, where $d$ is the dimension of the score and state representation. This history is converted into a surrogate target through an exponential *score tilt*, indexed with $\alpha$ that represents the *strength of repellence* in controlling the magnitude of the history-based repulsion. The surrogate family is normalization-free in the standard MCMC sense, yielding a generic wrapper: at each iteration, any base kernel targeting $\pi$ can instead be run on the current surrogate $\pi_{\theta_n}$ while the history is updated online. We analyze the coupled evolution of the history recursion and Monte Carlo estimators using stochastic approximation with controlled Markovian noise, establishing almost sure convergence and a joint central limit theorem. We further identify regimes in which the asymptotic covariance decreases as $\alpha$ increases, with scaling $O(1/\alpha)$, extending the near-zero-variance effect of finite-state history-dependent samplers to general state spaces with constant memory. Experiments on continuous targets and discrete energy-based models demonstrate improved estimator variance and mode coverage, while retaining $O(d)$ memory usage and modest per-iteration overhead.

Sparse Mixture of Experts (SMoE) models scale the capacity of models while maintaining constant computational overhead. SMoE methods fall into two categories: *Token Choice*, which routes each token to a fixed number of experts, and *Expert Choice*, which assigns a fixed number of tokens to each expert. However, the use of fixed budgets for tokens or experts causes both approaches to select irrelevant token–expert pairs or overlook critical assignments, which degrades overall performance. To fill that gap, we rethink SMoE from a *unified perspective* through the lens of *linear programming*, which provides a general formulation for SMoE models. Furthermore, we introduce **Unified Sparse Mixture of Experts (USMoE)**, a novel framework comprising a *unified mechanism* and a *unified score* to overcome these limitations. We provide both theoretical justification and empirical evidence demonstrating USMoE's effectiveness. Extensive evaluations across diverse data settings (clean and corrupted), multiple domains (including texts and vision tasks), and different learning approaches (training-free and training-based) show that USMoE not only delivers significant performance improvements over existing SMoE methods, but also enables more flexible expert selection budgets, reducing inference costs without compromising model performance.

High-dimensionality of data is often regarded as a fundamental statistical impediment in Machine Learning and AI. The purpose of this paper is to clarify, on the contrary, when and how high-dimensionality may be beneficial. In the setting of a general random function model of data we delineate between three notions of dimensionality: *effective dimension* $p_{\mathrm{eff}}$, measuring total variability across feature directions; *correlation rank* $r$, measuring functional complexity across samples; and *latent intrinsic dimension* $d$ of manifold structure hidden in data. Via a generalized Hanson-Wright inequality, we show that increasing $p_{\mathrm{eff}}$ drives a *blessing of dimensionality* phenomenon, whereby data dot-products concentrate about their expectations. In turn, we show that, under mild continuity assumptions (ensuring that features bring additional information as dimension grows), persistence diagrams recover latent homology when $p_{\mathrm{eff}} \in \omega (\log n)$ as $n\to\infty$. Informed by our theory, we revisit the ground-breaking neuroscience discovery of toroidal structure in grid-cell activity made by Gardner et al. (2022): our findings provide the first empirical evidence that this structure is *isometric* to a flat torus model of physical space, suggesting that grid cell activity conveys a geometrically faithful representation of the real world.

We study a variant of the stochastic multi-armed bandit problem in which the learner aims to identify and play an arbitrary arm whose expected reward exceeds a known satisficing threshold $S$, rather than optimizing against the best arm. Prior work has shown that when such a satisficing arm exists, time-independent bounds on the satisficing regret are achievable, but these guarantees deteriorate when an arm lies close to the threshold. We focus on instances in which the excess gap $\Delta_*$ (gap between the best arm and the threshold) is small relative to the suboptimality gaps $\Delta_i$, a regime that exposes this limitation. To capture this challenge, we introduce a refined notion of regret and propose a new algorithm, uncertain-UCB, which achieves *satisficing* pseudo-regret of $ O \left(\sum_{i: \Delta_i > \Delta_*} \frac{\ln(K/\Delta_*)}{\Delta_i}\right), $ while recovering standard pseudo-regret bounds when no arm exceeds the threshold. Further, we establish a near-matching lower bound in the small excess-gap regime, showing that any algorithm incurs at least $ \Omega \left(\sum_{i: \Delta_i > \Delta_*} \frac{\ln \big(\frac{\Delta}{(K-1) \Delta_* }\big)}{\Delta_i}\right) $ satisficing pseudo-regret.

Unsupervised Environment Design (UED) offers a promising paradigm for improving reinforcement learning generalization by adaptively shaping training environments, but it requires reliable environment evaluation to remain effective. However, existing UED methods evaluate environments using indirect proxy signals such as regret, value-based errors, or Monte Carlo, which suffer from bias, high variance, or substantial computational overhead. To address these limitations, we propose Parameter Change Environment Design (PACE), a general framework for adaptive level selection in UED. PACE evaluates a level by performing a provisional policy update on it and scoring it with the squared $\ell_2$ norm of the induced parameter change, which directly reflects realized learning progress. This score then guides level selection: levels enter a staleness-aware buffer based on their score, and are replayed via rank-based prioritization, inducing a curriculum that adapts to the agent’s evolving capability. By grounding environment evaluation in intrinsic optimization progress, PACE provides a low-variance evaluation signal and avoids the need for additional environment rollouts. Experiments on MiniGrid and Craftax demonstrate that PACE consistently outperforms established UED baselines in zero-shot generalization across diverse out-of-distribution evaluation protocols.

Predictive code completion greatly accelerates how quickly developers work. In spreadsheets, despite being much more common, such auto-completion features are virtually non-existent. To address this gap, we introduce a benchmark for systems that observe a sequence of user actions in a spreadsheet and predict future actions. Two challenges are (1) the absence of edit histories in public spreadsheet corpora and (2) the complex space of spreadsheet actions (spatial, temporal, composite). To address (1), we manually curate 52 sequences of 12K actions that recreate spreadsheets from public corpora, seeded by parametrized heuristics and LLM refinement. To address (2), we propose an online evaluation that expects a prediction after each user action, accepts or rejects that prediction, updates the future actions upon acceptance, and repeats this until the target spreadsheet is obtained. We use multiple baseline predictors (including zero-shot LLMs, fine-tuned SLMs, and classical models) and analyze different properties that our benchmark teaches us, including but not limited to: properties of saved actions and false positives, efficiency, effect of user profiles, effect of triggers, and effect of context.

Model merging offers an efficient solution for integrating task-specific knowledge from multiple fine-tuned models. Most existing approaches focus on manipulating the difference vectors between fine-tuned and pretrained weights, often overlooking the generalization capabilities inherent in the pretrained parameters. In this work, we revisit the role of pretrained weights in model merging and investigate their efficacy from a subspace perspective. We find that the components of pretrained weights residing in the core subspace—defined by the dominant singular vectors—are essential for maintaining generalization across diverse tasks. Specifically, we present empirical evidence that pretrained weights are nearly first-order stationary and exhibit predominantly non-negative curvature within this core subspace with respect to multi-task loss landscapes, indicating near-optimality. These findings suggest that task-specific adaptations should be injected primarily into the orthogonal complement of the core subspace, thereby preserving the generalization properties of the pretrained model. Extensive experiments on vision and vision-language tasks show that this subspace-aware strategy consistently yields improvements over state-of-the-art training-free merging methods, including Task Arithmetic, LOT Merging, ISO, and TSV. The source code is available at https://github.com/SunWenJu123/model-merging.

Estimating heterogeneous treatment effects is central to data-driven decision-making, yet industrial applications often face a fundamental tension between limited randomized controlled trial (RCT) budgets and abundant but biased observational data (OBS) collected under historical targeting policies. Although observational logs offer the advantage of scale, they may suffer from severe policy-induced imbalance and overlap violations, rendering standalone estimation unreliable. We propose a \textit{budgeted active experimentation} framework that iteratively collects informative randomized samples for causal effect estimation via active sampling. By leveraging observational signals, we develop an acquisition function targeting uplift estimation uncertainty, domain discrepancy, and overlap deficits to select the most informative units for randomized experiments. We establish finite-sample deviation bounds, asymptotic normality via martingale CLTs, and minimax lower bounds showing near-optimality in the linear representation setting. Experiments on synthetic datasets support our theoretical findings, and further extensions to industrial neural network-based uplift modeling scenarios show that active sampling can improve sample efficiency over random sampling under limited RCT budgets.

Differentiable approaches to causal discovery have shown promise in learning DAG structures via continuous optimization, but their theoretical guarantees are largely restricted to models with homoscedastic noise or known noise distribution. In particular, existing methods based on mean squared error fail to identify the true DAG when noise distributions are non-Gaussian and vary in scale. In this paper, we address this gap in nonlinear additive noise models (ANMs) with arbitrary noise. Our approach extends NOTIME (Berrevoets et al. 2025) which minimizes an independence criterion among the residuals. We show that the global minimizer of the independence criterion corresponds to the true underlying DAG up to additional constant edges in general ANMs. To recover the exact structure, we introduce an adaptive group lasso penalty that regularizes entire columns of the first-layer weight matrix of an MLP, enabling the selective pruning of constant edges in a functionally meaningful way. Empirically, our method achieves effective and stable performance across diverse noise types and variances, outperforming prior methods that lack identifiability guarantees in this setting.

Many organizations lack computational resources to fine-tune large language models (LLMs) on private (unshareable) data for better utility, while fine-tuning tiny language models (TinyLMs) alone performs poorly. To address this bottleneck, we propose a data-free knowledge distillation framework that generates LLM update vectors based on TinyLMs fine-tuned on private data. An update vector is a vector of parameter changes from an initial model to its fine-tuned version on a dataset, capturing the effect of cumulative gradient steps during fine-tuning. The key idea of our framework is a novel **Gradient Transformer** that transforms TinyLM's update vectors into LLM's update vectors. As derived from shadow datasets, $\texttt{Grad-Transformer}$ captures the correlation between TinyLM and LLM update vectors, enabling third-party providers to generate LLM update vectors given the organization's TinyLM update vectors without accessing the organization's private data. The framework supports multi-organization collaboration to jointly update LLMs, improving performance and cost-efficiency. Extensive experiments across language modeling and reasoning tasks show that $\texttt{Grad-Transformer}$ remarkably outperforms state-of-the-art knowledge distillation baselines, even under strict differential privacy protection.